Solving Parallel Lines in Coordinate Geometry: A Detailed Guide
When dealing with parallel lines in coordinate geometry, particularly on the Cartesian plane, it's essential to understand how the slope of a line is calculated and why parallel lines share the same slope. In this detailed guide, we will solve a specific problem involving parallel lines, namely: 'The line 3x6ky7 is parallel to the line 9kx8y5. What is k?'
Introduction to Parallel Lines in Coordinate Geometry
Parallel lines in coordinate geometry refer to lines that never intersect and have the same slope. This property helps us determine various characteristics of the lines such as their equations and the values that specific parameters must take for the lines to be parallel.
Understanding the Problem
The problem at hand involves two lines which are given in the form:
3x6ky 7
and
9kx8y 5
We need to determine the value(s) of k for which these lines are parallel. Let's break down the steps required to find this value.
Steps to Solve the Problem
Step 1: Rewrite the Equations in Slope-Intercept Form
To solve this problem, we will first convert both equations into the slope-intercept form, which is:
y ax b
In this form, 'a' represents the slope of the line, and 'b' is the y-intercept.
Step 2: Identify the Slopes of the Lines
For the first line, 3x6ky 7, we need to solve for y. We start by isolating y:
3x6ky 7
6ky 7 3x
y (-3x 7)/6k
slope (a1) -3/6k -1/(2k)
For the second line, 9kx8y 5, we also isolate y:
9kx8y 5
8y 5 9kx
y (5 9kx)/8
slope (a2) -9k/8
Step 3: Set the Slopes Equal to Each Other
Since the lines are parallel, their slopes must be equal:
-1/(2k) -9k/8
We can drop the negative sign and solve for k:
1/(2k) 9k/8
Multiplying both sides by 16k to eliminate the fractions:
8 18k2
18k2 8
k2 8/18 4/9
So, k ±2/3
Step 4: Verify the Solutions
Thus, the two possible values for k are:
k 2/3
k -2/3
To ensure these values are correct, we can substitute them back into the original equations and check if they result in parallel lines.
Conclusion
In conclusion, we have solved the problem to find the values of k for which the lines 3x6ky7 and 9kx8y5 are parallel. The solutions are k 2/3 and k -2/3.
Additional Resources
If you are further interested in coordinate geometry and parallel lines, consider exploring the following resources:
Math is Fun - Coordinate Geometry Khan Academy - Parallel Lines in Coordinate GeometryThese resources provide deeper insights and practice problems to further enhance your understanding of coordinate geometry and parallel lines.